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matter waves
Matthias Lopez, Jean-François Clément, Gabriel Lemarié, Dominique Delande, Pascal Szriftgiser, Jean Claude Garreau
To cite this version:
Matthias Lopez, Jean-François Clément, Gabriel Lemarié, Dominique Delande, Pascal Szriftgiser, et al.. Phase diagram of the Anderson transition with atomic matter waves. New Journal of Physics, Institute of Physics: Open Access Journals, 2013, 15, pp.065013. �10.1088/1367-2630/15/6/065013�.
�hal-00771284�
matter waves
M. Lopez1, J.-F. Clément1, G. Lemarié2,3, D. Delande3, P. Szriftgiser1, J. C. Garreau1,4
1Laboratoire de Physique des Lasers, Atomes et Molécules, Université Lille 1 Sciences et Technologies, CNRS; F-59655 Villeneuve d’Ascq Cedex, France
2Laboratoire de Physique Théorique UMR-5152, CNRS and Université de Toulouse, F-31062 France
3Laboratoire Kastler-Brossel, UPMC-Paris 6, ENS, CNRS; 4 Place Jussieu, F-75005 Paris, France
4Corresponding author
E-mail: jean-claude.garreau@univ-lille1.fr
Abstract. We realize experimentally a cold atom system equivalent to the 3D Anderson model of disordered solids where the anisotropy can be controlled by adjusting an experimentally accessible parameter. This allows us to study experimentally the disorder vs anisotropy phase diagram of the Anderson metal- insulator transition. Numerical and experimental data compare very well with each other and a theoretical analysis based on the self-consistent theory of localization correctly discribes the observed behavior, illustrating the flexibility of cold atom experiments for the study of transport phenomena in complex quantum systems.
PACS numbers: 03.75.-b , 64.70.Tg , 72.15.Rn
1. Introduction
The interplay of disorder and quantum interference has been an important subject in physics for more than 50 years. Quantum interferences, which are at the heart of most quantum effects, rely on precise relative phases between quantum trajectories, which are strongly sensitive to perturbations like decoherence (i.e. coupling with a large environment) and scattering of the wave function in potential wells. This last effect becomes particularly difficult to describe theoretically in a disordered system, in which these scattering processes have a random character. Intuitively, one easily understands that such kind of effect shall play an important role for example in the low-temperature electric conductance of solids. In fact, Anderson showed in 1958 that the presence of disorder might produce a spatial localization of the wavefunction, which suppresses conductivity [1] thus the name of “strong” localization.
Laser cooling opened the possibility of realizing very clean experiments in disordered systems, which generated a burst of interest on the subject. In adequate conditions, ultracold atoms placed in spatially structured laser beams feel this radiation as a mechanical potential acting on the center of mass variables of the atoms. Disordered potentials created in such a way allowed the realization of spatially disordered systems in one dimension [2, 3] and three dimensions [4, 5]. Despite these progresses, the Anderson metal-insulator transition (which manifests itself in 3 or more dimensions) is still very difficult to study in such systems, because Anderson localization requires a very strong disorder and – the cold atomic samples being prepared in the absence of disorder – the energy distribution of the atoms unavoidably spreads across the so- calledmobility edge, an energy threshold separating localized and extended eigenstates.
This in turn implies that the localized fraction, which can bedirectly measured in cold- atom experiments from the temporal evolution of the spatial probability distributions, remains small. Fortunately, one can find other systems also described by the Anderson localization physics, which are not a direct transposition of the condensed matter system, but rely on the profound analogy between quantum chaotic systems and disordered systems [6]. Using the quasiperiodic kicked rotor (QpKR) [7], an effectively 3D variant of the paradigmatic system of quantum chaos [8], the Anderson transition has been observed, its critical exponent measured experimentally [9, 10], its critical wavefunction characterized [11], and its class of universality firmly established [12], making this system an almost ideal environment to study Anderson type quantum phase transitions.
One advantage of this cold atom chaotic system as compared to other disordered systems is that the disorder can be controlled very precisely: the mean free path and the anisotropy are two experimentally tunable parameters. This allows us to present in this work an experimental study of the disorder vs anisotropy phase diagram of the Anderson transition, as well as an analytical description of these properties based on the self-consistent theory of Anderson localization, which brings another important brick to our detailed knowledge on the Anderson metal-insulator transition.
2. Controlled disorder and anisotropy within a cold atom system
The quasiperiodic kicked rotor is described by the one-dimensional time dependent Hamiltonian
Hqpkr = p2
2 +Kcosx(1 +ε cosω2t cosω3t)X
n
δ(t−n). (1) Experimentally, it is realized by placing laser-cooled atoms (of mass M) in a standing wave (formed by counterpropagating beams of intensity I0 and wavenumber kL) which generates an effective sinusoidal mechanical potential – nicknamed “optical potential”
– cosx acting on the center of mass position x of the atom. The standing wave is modulated by an acousto-optical modulator in order to form a periodic (of period T1) train of short square pulses whose duration τ is short enough that, at the time scale of atom center of mass dynamics, they can be assimilated to Dirac δ- functions. The amplitude of such pulses is further modulated with frequenciesω2 andω3, proportionally to 1 +ε cosω2t cosω3t. Lengths are measured in units of(2kL)−1, time in units of T1, momenta in units of M/2kLT1; note that [x, p] = i¯k with ¯k = 4~kL2T1/M playing the role of a reduced Planck constant. The pulse amplitude is K = ¯kτΩ2/8∆L, whereΩ is the resonance Rabi frequency between the atom and the laser light and ∆L the laser- atom detuning. Fixed parameters used throughout the present work are k¯ = 2.885, ω2 = 2π√
5,ω3 = 2π√ 13‡.
If ε = 0 one obtains the standard kicked rotor, which is known to display fully chaotic classical dynamics for K ≥ 6 [16]. At long time, the dynamics is a so-called chaotic diffusion in momentum space, which is – although perfectly deterministic – characterized by a diffusive increase of the r.m.s. momentum: hp(t)− p(t = 0)i = 0, h[p(t)−p(0)]2i ∼ 2Dt (which D the classical diffusion constant) where the average hi is performed over an ensemble of trajectories associated with neighbouring initial conditions. The statistical distribution ofp(t) has the characteristic Gaussian shape of a diffusion process, whose width increases like √
t. Quantum mechanically, this system displays the phenomenon of dynamical localization, that is, an asymptotic saturation of the average square momentum hp2i [8] at long time, that is localization in momentum space instead of chaotic diffusion, which has been proved to be a direct analog of the Anderson localization in one dimension [17, 18, 19].
If2π/T1, ω2, ω3,¯kare incommensurable andε6= 0one obtains the QpKR, which can be proven to be equivalent to the Anderson model in 3 dimensions [7, 10, 20, 21]. In a nutshell, the QpKR which is a 1-dimensional system with a time-dependent Hamiltonian depending on 3 different quasi-periods, can be mapped on a kicked “pseudo”-rotor, a 3- dimensional system with a time periodic Hamiltonian. As shown in detail in [10], both
‡ Rational values ofω2/2π, ω3/2πproduce a periodically – instead of quasiperiodically – kicked rotor, with different long time behaviour[13, 14, 15]. We chose “maximally irrational ratios” to avoid this problem.
systems share the same temporal dynamics. The Hamiltonian of the pseudo-rotor is:
H= p21
2 +ω2p2+ω3p3+Kcosx1[1 +εcosx2cosx3]X
n
δ(t−n), (2) with an initial condition taken as a planar source in momentum space (completely delocalized along the transverse directions p2 and p3). Note that the kinetic energy has a different dependence on the momentum in each direction: standard (quadratic) in direction 1, but linear in directions 2 and 3; hence, the name pseudo-rotor.
The Hamiltonian (2) is periodic in configuration space. It can thus be expanded in a discrete momentum basis composed of states|pi=|p= ¯kpi,where the pi are integers
§. In this basis, the evolution operator over one temporal period writes as the product U = JV of an on-site operator: V(p) = e−iφp with phases φp = ¯kp221 +ω2p2 +ω3p3 and of a hopping operator J such that:
hpf|J|pii=
Z dx (2π)3 exp
−iKcosx1(1 +εcosx2cosx3)
¯ k
exp
−i(pi−pf)x . (3) The phases V(p) are different on each site of the momentum lattice, and, although perfectly deterministic, constitute a pseudo-random sequence completely analogous to the true random on-site energies of the Anderson model. This makes it possible to identifyV as the disorder operator for the QpKR. The parameterKcontrols the hopping amplitudes, that is the transport properties in the absence of disorder. The larger K, the larger distance the system propagates in momentum space (with the operator J) before being scattered by the disorder operator V. As shown below, the associated mean free path in momentum space is of the order of K/¯k. Rather counter-intuitively, the weak disorder limit then corresponds to the largeK limit, that is strong pulses, while the strong disorder limit where Anderson localization is expected corresponds to small K. It should also be stressed that, for very small K, (very strong disorder), the system remains frozen close to its initial state, with a trivial on-site Anderson localization. This is not really surprizing as the classical dynamics is then regular instead of chaotic and even the classical chaotic diffusion is suppressed.
In the following, we will concentrate on the role of the ε parameter, which drives the anisotropy between the transverse directions and the longitudinal direction, showing the analogy of (2) with a system of weakly coupled disordered chains as considered in [22].
With such a system, we experimentally observed and characterized the Anderson transition [9, 10], which manifests itself by the fact that the momentum distribution is
§ This implies periodic boundary conditions. In general – especially for an ”unfolded“ rotor forx1 is a position in real space as realized in the experiment – one should use the Bloch theorem which guarantees the existence of states whose wavefunction take a phase factor after translation by2π.This amounts at considering not integerpi values, but ratherpi=ni+βi withni an integer andβi a fixed quantity called quasimomentum. All conclusions obtained in the simplest caseβi = 0 can be straightforwardly extended in the general case.
Figure 1.Schematic phase diagram of the metal-insulator Anderson transition for the quasi-periodic kicked rotor. The color plot corresponds to growth rateαofhp2(t)i ∝tα at long time (1000 kicks for this plot), estimated from numerical simulations. Blue color represents localization (α= 0), red represents diffusive dynamics (α= 1). The black line corresponds toα= 2/3,that is the critical line of the Anderson transition. Paths (white dashed lines) form the grid used for the determination ofKc(ε).
exponentially localizedΠ(p;t)∼exp (− |p|/ploc)(with ploc the localization length) ifK is smaller than a critical value Kc(ε) and Gaussian diffusive Π(p;t) ∼ exp (−p2/4Dt) (where D is the diffusion coefficient) for K > Kc(ε) after a sufficiently long time. At criticality, K =Kc(ε), the localization length diverges, the diffusion constant vanishes, and the critical state is found [11] to have a characteristic Airy shape
Π(p;t)≈ 3 2
α
pΛc(ε)t2/3Ai
"
α s
|p|2 Λc(ε)t2/3
#
(4) following the anomalous diffusion at criticality: hp2i = Λc(ε)t2/3 (here α = 31/6Γ(2/3)−1/2). The fundamental quantities characterizing the threshold of the transition are therefore Kc(ε) and Λc(ε), and we will consider in the following their dependence as a function of the anisotropy parameter ε.
3. Experimental determination of the anisotropy phase diagram
In our experience, we measure the population of the zero velocity class Π(0;t) using Raman velocimetry [24, 25, 26]. This quantity is proportional to hp2(t)i, with a
-2 -1.5 -1 -0.5 0
ln(ξ/t1/3)
1 2 3
ln(Λ)
(a)
5 6 7 8
K 0
1 2 3 4
ξ(K)
(b)
-1.5 -1 -0.5 0 0.5
ln(ξ/t1/3)
0 0.5 1 1.5 2 2.5
ln(Λ)
(c)
3 4 5 6
K 0
2 4 6 8
ξ(K)
(d)
Figure 2. Determination of the critical point from experimental data at two different anisotropyε. The finite-time scaling method (see text) applied to the experimental data Λ(t) ∝Π(p = 0;t)−2t−2/3 allows for a determination of the scaling function F (Eq. (6)) represented in a and c and the scaling parameterξ(K)shown in b and d. The critical point corresponds to the tip at the right of the scaling function (see a and c), at the intersection of the diffusive (top) and localized branch (bottom). The marked maximum ofξ(K)gives a precise determination ofKc. The parameters are: ε= 0.4 for a and b;ε= 0.8for c and d. tvaries up to120kicks.
ε K1−K2 Kc (exp) Kc (num) ln Λc(exp) ln Λc(num) 0.2 7.0-14.0 8.85 ± 0.1 8.84 ±0.47 2.1 ± 0.1 2.71 ± 0.44 0.3 5.2-9.2 7.46 ±0.05 7.71 ±0.42 2.05 ± 0.08 2.22 ± 0.34 0.4 4.5-8.5 6.75 ±0.04 6.77 ±0.52 1.95 ± 0.05 1.81 ± 0.47 0.5 4.0-8.0 6.00 ±0.04 5.93 ±0.37 1.85 ± 0.05 1.36 ± 0.46 0.6 3.4-7.4 5.59 ±0.04 5.27 ±0.35 1.75 ± 0.05 1.10 ± 0.30 0.7 2.9-6.9 5.27 ±0.03 4.99 ±0.34 1.60 ± 0.05 0.94 ± 0.40 0.8 2.8-6.8 4.84 ±0.03 4.70 ±0.43 1.52 ± 0.04 0.98 ± 0.31
Table 1. Experimental results on the determination of the critical point of the metal-insulator Anderson transition, for various values of the parameterεof the quasi- periodic kicked rotor. The second column indicates the range ofKwhere data has been taken. The experimentally measured values of bothKc andΛc are compared to the numerically calculated values. The uncertainties on the experimental data are rather small. The numerical data (with times up to1000kicks) have similar uncertainties, but also incorporatesystematic shifts of(Kc,Λc)as a function of time [23] which cannot be estimated in the experimental data due to the restricted range of observation times t≤120kicks.
0.2 0.3 0.4 0.5 0.6 0.7 0.8
ε
4 5 6 7 8 9 10 11
K c
(a)
0.2 0.3 0.4 0.5 0.6 0.7 0.8
ε
0.5 1 1.5 2 2.5 3
ln Λ c
(b)
Figure 3. (a) Position of the critical pointKc(ε), and (b) value of the criticalΛc(ε).
Numerical results (black diamonds) and experimental measurements (red circles) are represented with their associated error bars. The uncertainties on the experimental data are rather small, as can be directly seen in fig. 2. The numerical data (with times up to 1000 kicks) have similar uncertainties, but also incorporate systematic deviations of(Kc,Λc)when estimated over various temporal ranges. These systematic deviations cannot be easily measured in the experiment (limited to120kicks). In plot (a) one observes a very good agreement between numerical and experimental results.
The agreement is good in plot (b), except in the region of lowεwhere decoherence is expected to have a significant impact on the results and in the region of largeεwhere the finite variance of the initial momentum distribution tends to increase Λc at 120 kicks but has only small effect onKc.
proportionality factor which depends on the specific shape of Π(p). This factor varies smoothly across the Anderson transition, so that the transition can be studied using eitherhp2(t)iorΠ(0;t). The scaling theory of localization [27, 10] predicts that hp2ihas characteristic asymptotic behaviors in tα, with α = 0 in the localized regime, α = 2/3 in the critical regime, and α = 1 in the diffusive regime. This prediction has been fully confirmed by the experimental observations [11]. One can then define the scaling variable [9, 10]:
Λ(t)≡ hp2(t)i
t2/3 ∝ 1
Π(p= 0;t)2 t2/3 . (5) Asymptotically, Λ(t) ∝ t−2/3, t0, t1/3 in the localized, critical and diffusive regimes, respectively, so that ln Λ(t) vs lnt1/3 displays a positive slope 1 in the diffusive regime, zero slope at the critical point and negative slope−2in the localized regime, which allows one to unambiguously identify the critical point. However, experimental limitations prevent us from performing measurements at large enough times (in our experiments typically tmax = 120) to distinguish precisely between the localized and diffusive behaviors in the vicinity of the transition k. The main causes of this limitation is the falling (under gravity action) of the cold atoms out of the standing wave and decoherence
k Note however that for the parameters used here,tmax/tloc∼10wheretlocis the localization time for the lowestK value used in each series at fixed ε, so that thoe lowest point is clearly in the localized regime.
induced by spontaneous emission.
Fortunately, a technique known as finite size scaling (which is finitetime scaling in our case), based on arguments derived from the so-called one parameter scaling theory of the Anderson transition [27] allows us to overcome this limitation. The application of this technique to our problem has been discussed in details in previous works [10, 20, 28];
let us just say here that it relies on the verified hypothesis that Λ can be written as a one-parameter scaling function:
Λ =F
ξ(K) t1/3
, (6)
with the scaling parameterξ(K)which plays the role of the localization lengthplocin the localized regime and of the inverse of the diffusive constant in the diffusive regime. This method produces a rather precise determination of the critical parameters Kc(ε) and Λc(ε)and of the critical exponent of the Anderson transition [12] even from experimental signals limited to a hundred of kicks or so. An example of such a determination is presented in Fig. 2. The critical point corresponds to the tip at the right of the curve in Fig. 2a and c, at the intersection of the two clearly defined branches: A diffusive (top) and a localized branch (bottom). By construction, in principle ξ(K) should diverge at the critical point, but the finite duration of the experiment and decoherence effects produce a cutoff; however, it still presents, as shown in Fig. 2b and d a marked maximum at the transition, and a careful fitting procedure taking these effects into account [12]
allows a precise determination of Kc. Once the value of Kc has been determined according to the above technique, we measure the full momentum distribution, which is found to be in excellent agreement with the predicted Airy shape, eq. (4), as shown in [11]. A simple fit of the experimental data by an Airy function allows to measure hp2i, hence Λc.
We have measured the value of the critical parameters Kc(ε) and Λc(ε) [Eq. (5)]
for a grid of 7 paths at constant ε in the parameter plane (K, ε) (see Fig. 1) For each path, 50 uniformly spaced values of K are used and the values of Π(0;t) measured for each K value; the initial and final values orK are chosen symmetrically with respect to the critical point. For each value of K, an average of 20 independent measurements is performed, a full path thus corresponds to more than seven hours of data acquisition.
Table 1 gives the details of each path and the results obtained.
Figure 3 displays the experimental and numerical results. Plot (a) indicates the position of the critical point Kc(ε) and plot (b) the critical value Λc(ε). In both plots, experimental measurements are indicated by red circles, numerical simulation results by black diamonds and are represented along with their error bars. The uncertainty of the numerical data (see the following for a discussion of the numerical method) is evaluated from data up to t = 1000 kicks and thus incorporates systematic shifts of (Kc,Λc) as a function of time [23] which cannot be estimated in the experimental data due to the restricted range of observation times t ≤ 120 kicks. This results in larger numerical error bars than experimental ones. Note also that a small uncertainty in Kc
implies a much larger error in Λc due to its rapid variation as a function of K. In plot (a), one observes a very good agreement between numerical and experimental results.
In plot (b), the agreement is good, except in the region of low ε which corresponds to high values of K and are thus more sensitive to decoherence effects. In the region of large ε, the finite variance of the initial momentum distribution tends to increase the experimental Λc, an effect which is not present in the numerical data.
4. Self-consistent theory of the anisotropy phase diagram
We shall now try to describe theoretically the observed anisotropy dependences of the two critical parameters Kc(ε) and Λc(ε). The approach we shall follow is based on the self-consistent theory of localization [29] which has been used successfully to predict numerous properties of the Anderson transition, and in particular the disorder vs energy [30] and disorder vs anisotropy [22] phase diagrams of the 3D Anderson model.
Moreover, the self-consistent theory of localization has been transposed to the case of the kicked rotor [31, 32]. We will use in the following a simple generalization of this latter approach adapted to the case of the 3D anisotropic kicked pseudo-rotor (2) corresponding to the quasiperiodic 1D kicked rotor.
The starting point is to consider the probability to go from a site pi to a site pf in N steps, P(pi,pf, t = N) ≡ |hpf|UN|pii|2. It consists in propagations mediated by the hopping amplitudes hpn+1|J|pni and by collisions on the disorder represented by V(p) = e−iφp. Two important points are the following: (i) one can consider in a first approximation the φp as completely random phases [17, 33, 34] and we will consider quantities averaged over those phases, for example P(pi,pf, t =N) where the line over the quantity represents this averaging; (ii) hpn+1|J|pni plays the role of the disorder averaged Green’s function (in the usual language of diagrammatic theory of transport in disordered systems [35]), that is the propagation between two scattering events. Indeed, when ε= 0 and in the direction p1, this is just a Bessel function which decreases exponentially fast for |pn+1−pn| ≫ K/¯k and one can thus see K/¯k as the analog of the mean free path, with the limit of small disorder corresponding toK/¯k ≫1.
One can attack the problem of the calculation of P by looking for propagation terms – including of course interference patterns – which survive the disorder averaging.
At lowest order, the contribution containing no interference term to the probability P is called the Diffuson [35]. It corresponds to the classical chaotic diffusion and can be shown to have a diffusive kernel expressed in the reciprocal space (q, ω)(conjugated to (p, t)) as [31, 32, 20]:
PD(q, ω) = 1
−iω+P
jDjjqj2 . (7)
Here, the diffusive tensorD– computed in [36] for largeK– is anisotropic, but diagonal,
with:
D11 = K2 4¯k2
1 + ε2
4
, D22 =D33 = K2ε2
16 ¯k2 .
(8) This anisotropic diffusive kernel is valid at long times and on large scale in momentum space, that is in the so-called hydrodynamic limitω ≪1andqjkj ≪1, withkj the mean free path along direction j which is such that Djj =kj2/4. Equation (7) means that in the regime of long times and large distances (in momentum space), we should have a diffusive transport with hp2ji= 2Djjt. This is certainly not the case near the Anderson transition, which implies that we must go beyond the Diffuson approximation.
The simplest interferential correction – known as weak localization correction – is due to the constructive interference between pairs of time-reversed paths ¶, or equivalently to the maximally crossed diagrams, or Cooperon. The net effect of these interferential contributions is to increase the return probability at the initial point and to decrease the diffusion constant. It is possible to compute perturbatively the weak localization correction as an integral (see below) depending on the diffusion constant itself. Contributions from higher orders are extremely complicated and there is no systematic way of summing them all.
The self-consistent theory of localization is a simple attempt at approximately summing the most important contributions: instead of computing the weak localization correction using the raw diffusion constant, one uses the one renormalized because of weak localization. The whole thing must of course be self-consistent, so that the diffusion constant computed taking into account the weak localization correction is equal to the one input in the calculation of this correction. The price to pay is that one can no longer define a single diffusion constant – or, in the anistropic case, a single diffusion tensor – but must introduce a frequency-dependent diffusion constant (or diffusion tensor).
This is nevertheless quite natural if one wants to describe the transition from a diffusive behaviour as short time (large frequency) when interference terms are small to a localized behaviour at long time (small frequency). The intensity propagator P takes then the approximate form: P(q, ω) = −iω+P1
jDjj(ω)qj2 with the frequency dependent diffusion constant following the self-consistent equation [20, 29, 31]:
Dii(ω) = Dii−2Dii(ω)
Z d3q (2π)3
1
−iω+P
jDjj(ω)qj2 . (9)
Quite remarkably, this approach is able to account for a certain number of observed features: it predicts a transition between a metallic phase of diffusive transport for K > Kc(ε) where Dii(ω) ∼
ω→0 Dii(0) > 0, to a localized phase for K < Kc(ε) where Dii(ω) ∼
ω→0−iωploc2
i withplocithe localization length along directioni. At the threshold,
¶ The quasi-periodic kicked rotor or the equivalent periodic 3D pseudo-rotor is not invariant by time reversal. However, the Hamiltonian, eq. (2), is invariant by the product of time reversal and parity.
The existence of such an anti-unitary symmetry is sufficient for the system to belong to the Orthogonal universality class, and consequently for the existence of the weak localization correction.
the transport is predicted to follow an anomalous diffusion with Dii(ω)∼(−iω)1/3 and this implies the Airy shape of the critical state observed experimentally [11]. In the following, we will calculate explicitly the critical parameters Kc(ε) and Λc(ε) from (9) and show that it also complies well with the experimental observations.
One shall evaluate the integral on the right hand side of equation (9). It is important to remark that, although the system is anistropic and thus 3 different equations (9) have to be solved simultaneously, they in fact follow exactly the same renormalization scheme:
dividing eq. (9) by Dii produces the same equation in all 3 dimensions. In other words, there is no anomaly in the anisotropic character at the critical point.
It is well known [29] that in dimension d ≥ 2, the results of the self-consistent theory are cutoff dependent. Indeed, the integral in (9) diverges at largeq and must be limited to qj < qmaxj , where qjmax is a cutoff on the order of k−1j , i.e. the inverse of the mean free path. In the following, we will take [22] qjmax ≡ C1k−1j = C1/(2p
Djj) with C1 a numerical constant of the order of one. We make the following change of variables:
Yj =q
Djj(ω)
−iω qj and define the rescaled cutoff: ℓ(ω) ≡ q
Djj(ω)
−iω qjmax (from Eq. (9) it is clear that the ratio Djj(ω)/Djj is isotropic, thus ℓ(ω)is isotropic). One obtains:
Dii(ω)
Dii = 1− C1
2π2
√ 1
D11D22D33
1− tan−1ℓ(ω) ℓ(ω)
. (10)
The threshold Kc(ε) of the Anderson transition is then approached from the diffusive regime, which is caracterized by DDii(ω)
ii ω→0→ DDii(0)ii →
K→Kc
0 and ℓ(ω) ω→0→ ∞. Therefore, Kc(ε)is such that:
D11D22D33 = C12
4π4 . (11)
From the above expressions (8) for the diffusion tensor, one deduces the following dependence of the threshold vs anisotropy:
Kc(ε) =
24C1
π2 1/3
¯ k (ε2p
1 +ε2/4)1/3 . (12) The self-consistent theory allows also for a determination of Λc(ε). In fact, at finite but sufficiently small ω (i.e. at sufficiently large times), ℓ(ω)is large and one can evaluate the right hand side of (10) at the lowest order in 1/ℓ(ω) which gives:
Dii(ω)
(−iω)1/3 = 1 (2π)2/3
D11
(D11D22D33)1/3 . (13) We know from the study of the critical state of the Anderson transition [11] that
D11(ω)
(−iω)1/3 = Γ(2/3)3 Λc which allows us to write:
Λc(ε) = 3 Γ(2/3)
1 2π
2/3 D211 D22D33
1/3
. (14)
Using the diffusion tensor relations (8), one obtains finally:
Λc(ε) = 3 Γ(2/3)
2 π
2/3
1 +ε2/4 ε2
2/3
. (15)
Equations (12) and (15) are the most important results of this section. They predict that the threshold Kc(ε) and the critical anomalous diffusion parameter Λc(ε) diverge at large anisotropy as Kc(ε) ∼ ε−2/3 and Λc(ε) ∼ ε−4/3. Therefore, when ε = 0 we recover the case of the 1D periodic kicked rotor which is always localized whatever the kicking amplitude K.
5. Experimental and numerical tests of the self-consistent theory of the anisotropy phase diagram
In order to test the predictions of the self-consistent theory, we have performed numerical simulations of the dynamics of the quasi-periodic kicked rotor (1). We have determined the critical parameters Kc(ε) and Λc(ε) from the crossing of the curvesln Λ = lnhpt2/23i vs K at different times (see figure 4). At the critical point,Λ(t)is a constant Λ(t) = Λc(ε) corresponding to the critical anomalous diffusion hp2i ∼ t2/3 and the crossing of the curves in fig. 4, whereas for K < Kc(ε) (K > Kc(ε), resp.), Λ(t) decreases (increases, resp.) as time increases. We have evaluated the uncertainty of the parameters (Kc,Λc) by the region where the evolution ofΛ(t)is not monotonous (due to systematic shifts of (Kc,Λc)as a function of time [23]). The results are represented in fig. 5 forKc(ε)and in fig. 6 forΛc(ε)by the white filled region between the blue filled region where the system is localized and pink filled region where the dynamics is diffusive. The data seem to follow an algebraic increase as the anisotropy parameter ε decreases (linear dependence in log-log scale as in figs. 5 and 6), on top of which oscillations are also clearly seen.
The self-consistent theory discussed in the previous section predicts that the critical regime of the Anderson transition is given by Eq. (11). Various approximations – leading to slightly different predictions for the position of the critical point – can be used for the values of the components of the diffusion tensor:
• (i) The simplest approximation is to use eq. (8), valid asymptotically for large K. This results in the simple analytic predictions (12) and (15), represented by black lines in figures 5 and 6. The algebraic dependences of Kc(ε) and Λc(ε) are well accounted for by these simple predictions, however they fail to reproduce the oscillating corrections observed in the numerical data.
• (ii) The theoretical prediction (8) for the diffusion tensorD miss the oscillations of the diffusion tensor of the 3D kicked pseudo-rotor (2) vsK and¯k. Such oscillations are well known in the case of the 1D periodic kicked rotor [37, 33] and arise due to subtle temporal correlation effects. In our case, we have checked that they could be described approximately by the known oscillating form [33], but only along direction
23 24 25 26 27 28 29 4
5
ln Λ
23 24 25 26 27 28 29
K
0 1 2 3 4
16π4 D 11D 22D 33
Figure 4. Method of determination of the thresholdKc(ε). Upper panel: Numerical data forln Λ = lnhp
2i
t2/3 vsKat different times ranging fromt= 36to956. The threshold corresponds to the crossing of these lines where we have the critical anomalous diffusion hp2i ∼ t2/3, whereas for K < Kc(ε), Λ(t) decreases at long time (localized regime) and for K > Kc(ε) Λ(t) increases at long time (metallic regime). The uncertainty region (between the orange and violet dash-dotted lines) corresponds to the region where the evolution ofΛ(t)is not monotonous. Lower panel: The different degrees of approximation for16π4D11D22D33. According to Eq. (11) (with C1 = 1/2, see text), this quantity should be equal to unity at the critical point. The black line corresponds to the simple analytic prediction (8) for the diffusion tensor. The red line shows the theoretical prediction incorporating oscillating corrections for the diffusion tensor (see Eq. (16) and text). The green line with points shows numerical data for the diffusion tensor D of the 3D kicked rotor (2), at short time. The parameters are: ¯k = 2.89, ω2= 2π√
5 andω3= 2π√
13,ε= 0.036.
1: D˜11≈D11× {1−2J2( ˜K)[1−J2( ˜K)]},
D˜22= ˜D33 =D22=D33. (16) with K˜ ≡Ksin ¯¯k/2k/2 and J2 the usual Bessel function. The use of the above equation for the diffusion tensor allows for a better analytical description of the anisotropy phase diagram, see the red lines in figures 5 and 6.
• (iii) The third type of approximation relies on a direct numerical calculation of the diffusion tensor D of the 3D kicked rotor (2) at short time by a linear fitting procedure over the first ten kicks ofhp2ii = 2Diit. This gives a numerical prediction of the self-consistent theory for the anisotropy phase diagram represented by the
0.02 0.05 0.10 0.20 0.50 1.00 5
10 20 50
ε Kc
Figure 5. Threshold of the Anderson transition vs anisotropy (log-log scale). The anisotropy dependence of Kc(ε) with the associated uncertainty is determined from numerical simulations of the dynamics of the quasi-periodic kicked rotor and is represented by the white filled region between the blue (localized) and pink (metallic) filled regions. The three degrees of approximation of the self-consistent theory prediction (11) (with C1 = 1/2) are shown: (i) the black line corresponds to the simple analytic prediction (12), (ii) the red line incorporates oscillating corrections for the diffusion tensor (see Eq. (16) and text) while (iii) the green line with points corresponds to numerical data for the diffusion tensorD of the 3D kicked rotor (2), at short time. The blue points represent the experimental data. The parameters are:
¯
k= 2.89,ω2= 2π√
5 andω3= 2π√ 13.
0.02 0.05 0.10 0.20 0.50 1.00 5
10 50 100 500 1000
ε Λc
Figure 6. Critical parameter Λc vs anisotropy (log-log scale). The anisotropy dependence of Λc(ε) with the associated uncertainty is determined from numerical simulations of the dynamics of the quasi-periodic kicked rotor and is represented by the white filled region between the blue (localized) and pink (metallic) filled regions.
The prediction (14) of the self-consistent theory is shown with the three different degrees of approximations considered: (i) the simple analytic prediction (15) is shown in black line, (ii) the red lines is based on the analytic prediction Eq. (16) for the diffusion tensor incorporating oscillating corrections in K and ¯k and (iii) numerical data for the diffusion tensorD of the 3D kicked rotor at short time (2) give the green line with points. The blue points represent the experimental data. The parameters are: ¯k= 2.89,ω2= 2π√
5 andω3= 2π√ 13.
